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Helicoidal surfaces with prescribed curvatures in \(\mathrm{Nil}_3\). (English) Zbl 1284.53058

Summary: We study helicoidal surfaces in the 3-dimensional Heisenberg group \(\mathrm{Nil}_3\). Also, we construct helicoidal surfaces in \(\mathrm{Nil}_3\) with prescribed Gaussian curvature or mean curvature given by smooth functions. As result, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C30 Differential geometry of homogeneous manifolds
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